Electronic – Mosfet behaving differently with other power supply and same voltage

1 - The LED strip says that it needs 12 volts at 3 amps. You are providing 12 volts at 1 amp. I'm tempted to ask, "Do you see a problem?" but you obviously don't. So get at least a 3 amp supply, already.

2 - You are driving the transistor bases directly, rather than through a base resistor. This is causing the Arduino to try to provide way more current than it can, so the Arduino is getting hot.

3 - If you do provide a base resistor which makes the Arduino happy, the transistors will not be driven hard enough to work without a heat sink, and they will get very hot and probably fail if you do it long enough.

4 - Regardless of whether the transistors fail, they will not be able to drive the LEDs to much brightness. A single transistor will not have enough gain. Sergei Grishin has the right idea, but make sure that the MOSFET can drive one amp reliably (some cannot, although most can), and that the maximum "gate threshold voltage" - Vgss - is greater than the output voltage of the Arduino pins. This will not be a problem if your Arduino is putting out 5 volts, but some MOSFETs will not work reliably with 3.3 volts on their gates.

A single transistor will not provide an interface with your FET AND do it without inversion. You can do the job with two transistors, though.

^{simulate this circuit – Schematic created using CircuitLab}

This will provide a decent drive for your FET with about a 1 usec added to on-time. That is, if you put in a 10 usec pulse, the FET will be on for about 11 usec.

Also note that R5 needs to be a 1-watt resistor.

EDIT -

It turns out I was using the wrong FET model, and my value for gate capacitance was too low. For a "real" IRFZ44, the added pulse width is about 3 usec, not 1. The problem is that, when you go to turn off the FET, The charge stored in the gate can only discharge through R5, and the time constant is in the 3 usec range. You can, of course, decrease R5, but then the power it dissipates when the load is driven on goes up. You can get a 2 usec delay rather than 3 by decreasing R5 to 100 ohms, but then you need a 2-watt resistor. If you need very narrow PWM cycles (low effective motor current) you're probably better off with a more sophisticated driver. I personally tend to go with the Maxim MAX4426/4427 (less than $4 each, with 2 channels per IC), but that's just habit.